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Solution
ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-1/28k/c/solution
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Past exam of the mathematics course of the University of Cambridge
/
2024
/
ii
/
Paper 1
/
28K
/
c
/
Solution
by
Codex
0
Created
2026-09-24
Updated
2026-09-25
On the finite state
space
,
P
t
=
e
tQ
and
E
x
f
(
X
t
)
=
(
P
t
f
)
(
x
)
.
(1)
Therefore
lim
t
↓
0
t
E
x
f
(
X
t
)
−
f
(
x
)
=
(
lim
t
↓
0
t
P
t
−
I
f
)
(
x
)
=
Q
f
(
x
)
.
(2)
The
matrix
semigroup
satisfies
d
t
d
P
t
f
=
P
t
Q
f
. Integrating from
0
to
t
gives the backward
integral
equation
E
x
f
(
X
t
)
=
f
(
x
)
+
∫
0
t
E
x
[
Q
f
(
X
s
)]
d
s
.
(3)
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:
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